QUANTUPLICITY: RATIO, MEASURE, AND PROPORTION

Diagrams

The following diagrams are numbered to follow the text at the left.

Example 1

A and B are two magnitudes where A is larger than B. How much larger?

We set B alongside A and count off the number of times we use B to match A (its quantuplicity). Here B is used twice.

We state the comparison as a ratio. Here we show the comparison of the two magnitudes, then write the ratio with their names.

Example 2

This gives a ratio:

PROPORTION

  This is a proportion. Notice that it has two ratios and four terms (the four shapes). We can write the proportion with letters:



We can then count the quantuplicity of the squares and write the ratio that appears on both sides of the '=' sign in the proportion:

Properties of proportion.

Reciprocal Property
Now if we have this proportion:

Can we also write this proportion? Note that we have inverted the ratios. That is why this is called the "reciprocal property."

Yes. Since the ratios are the same on both sides, the proportion holds. In other words, if

then

**Reciprocal Property**
Now if we have this proportion:
Can we also write this proportion? Note that we have inverted the ratios. That is why this is called the "reciprocal property."
Yes. Since the ratios are the same on both sides, the proportion holds. In other words, if then

Exchange Property

Again, if we have this proportion:

Can we also write this proportion? Look carefully and you will see that the two brown squares measure the two red squares the same number of times as the four green squares measure the blue square.

Yes. Since the ratios are the same on both sides, the proportion holds. In other words,

if

then

Here we have "exchanged" green and red (X and Y). Hence the "exchange property."

**Exchange Property**
Again, if we have this proportion:
Can we also write this proportion? Look carefully and you will see that the two brown squares measure the two red squares the same number of times as the four green squares measure the blue square.
Yes. Since the ratios are the same on both sides, the proportion holds. In other words, if then Here we have "exchanged" green and red (X and Y). Hence the "exchange property."

Cross-multiplication

Again, our original proportion:Notice that this time I have added lines so that we can measure all the shapes by the smallest brown square.

Suppose we were to multiply the number of small blue squares (W) by the number of brown squares (Z) ?

Count and you will see that it yields
16 x 2 = 32.

Now multiply the red squares (X) by the number of green squares (Y): 8 x 4 = 32.

So by the transitive property of equality, 16 x 2 = 8 x 4.

Since the two results are the same (32 squares), we can say that

if

then

This is the well-known "cross-multiplication."

**Cross-multiplication**
Again, our original proportion:Notice that this time I have added lines so that we can measure all the shapes by the smallest brown square.
Suppose we were to multiply the number of small blue squares (W) by the number of brown squares (Z) ? Count and you will see that it yields 16 x 2 = 32. Now multiply the red squares (X) by the number of green squares (Y): 8 x 4 = 32. So by the transitive property of equality, 16 x 2 = 8 x 4.
Since the two results are the same (32 squares), we can say that if then This is the well-known "cross-multiplication."

We can write the ratios of the sides as a proportion and then write the proportion in terms of measures (numbers).